The optimal power flow (OPF) problem seeks to control power generation/demand to optimize certain objectives such as minimizing the generation cost or power loss in the network. It is becoming increasingly important for distribution networks, which are tree networks, due to the emergence of distributed generation and controllable loads. In this paper, we study the OPF problem in tree networks. The OPF problem is nonconvex. We prove that after a "small" modification to the OPF problem, its global optimum can be recovered via a second-order cone programming (SOCP) relaxation, under a "mild" condition that can be checked apriori. Empirical studies justify that the modification to OPF is "small" and that the "mild" condition holds for the IEEE 13-bus distribution network and two real-world networks with high penetration of distributed generation.
We study convex relaxations of nonconvex quadratic programs. We identify a family of so-called feasibility preserving convex relaxations, which includes the well-known copositive and doubly nonnegative relaxations, with the property that the convex relaxation is feasible if and only if the nonconvex quadratic program is feasible. We observe that each convex relaxation in this family implicitly induces a convex underestimator of the objective function on the feasible region of the quadratic program. This alternative perspective on convex relaxations enables us to establish several useful proper…
This work addresses the occupation measure relaxation of calculus of variations problems, which is an infinite-dimensional linear programming relaxation amenable to numerical approximation by a hierarchy of semidefinite optimization problems. We address the problem of equivalence of this relaxation to the original problem. Our main result provides sufficient conditions for this equivalence. These conditions, revolving around the convexity of the data, are simple and apply in very general settings that may be of arbitrary dimensions and may include pointwise and integral constraints, thereby co…
Non-convex optimization problems can be approximately solved via relaxation or local algorithms. For many practical problems such as optimal power flow (OPF) problems, both approaches tend to succeed in the sense that relaxation is usually exact and local algorithms usually converge to a global optimum. In this paper, we study conditions which are sufficient or necessary for such non-convex problems to simultaneously have exact relaxation and no spurious local optima. Those conditions help us explain the widespread empirical experience that local algorithms for OPF problems often work extremel…
This paper proposes a general fixture layout design framework that directly integrates the system equation with the convex relaxation method. Note that the optimal fixture design problem is a large-scale combinatorial optimization problem, we relax it to a convex semidefinite programming (SDP) problem by adopting sparse learning and SDP relaxation techniques. It can be solved efficiently by existing convex optimization algorithms and thus generates a near-optimal fixture layout. A real case study in the half-to-half fuselage assembly process indicates the superiority of our proposed algorithm …
We consider the global optimization of nonconvex mixed-integer quadratic programs with linear equality constraints. In particular, we present a new class of convex quadratic relaxations which are derived via quadratic cuts. To construct these quadratic cuts, we solve a separation problem involving a linear matrix inequality with a special structure that allows the use of specialized solution algorithms. Our quadratic cuts are nonconvex, but define a convex feasible set when intersected with the equality constraints. We show that our relaxations are an outer-approximation of a semi-infinite con…
This paper aims to find efficient solutions to a multi-objective optimization problem (MP) with convex polynomial data. To this end, a hybrid method, which allows us to transform problem (MP) into a scalar convex polynomial optimization problem (P$_z$) and does not destroy the properties of convexity, is considered. First, we show an existence result for efficient solutions to problem (MP) under some mild assumption. Then, for problem (P$_z$), we establish two kinds of representations of non-negativity of convex polynomials over convex semi-algebraic sets, and propose two kinds of finite conve…
In this paper we introduce the Ameso optimization problem, a special class of discrete optimization problems. We establish its basic properties and investigate the relation between Ameso optimization and the convex optimization. Further, we design an algorithm to solve a multi-dimensional Ameso problem by solving a sequence of one-dimensional Ameso problems. Finally, we demonstrate how the knapsack problem can be solved using the Ameso optimization framework.
In this paper, we consider the optimal power flow (OPF) problem which consists in determining the power production at each bus of an electric network by minimizing the production cost. Our contribution is an exact solution algorithm for the OPF problem. It consits in a spatial branch-and-bound algorithm based on a compact quadratically constrained convex relaxation. This compact relaxation is computed by solving the rank relaxation once at the beginning of the algorithm. The key point of this approach is that the lower bound at the root node of the branch-and-bound tree is equal to the rank re…
In this paper, we study the mixed-integer nonlinear set given by a separable quadratic constraint on continuous variables, where each continuous variable is controlled by an additional indicator. This set occurs pervasively in optimization problems with uncertainty and in machine learning. We show that optimization over this set is NP-hard. Despite this negative result, we characterize the structure of the convex hull, and show that it can be formally studied using polyhedral theory. Moreover, we show that although perspective relaxation in the literature for this set fails to match the struct…
Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromi…
Given a nonlinear, univariate, bounded, and differentiable function $f(x)$, this article develops a sequence of Mixed Integer Linear Programming (MILP) and Linear Programming (LP) relaxations that converge to the graph of $f(x)$ and its convex hull, respectively. Theoretical convergence of the sequence of relaxations to the graph of the function and its convex hull is established. For nonlinear non-convex optimization problems, the relaxations presented in this article can be used to construct tight MILP and LP relaxations. These MILP and the LP relaxations can also be used with MILP-based and…
Flexible transmission line impedances on one hand are a promising control resource for facilitating grid flexibility, but on the other hand add much complexity to the concerned optimization problems. This paper develops a convexification method for the AC optimal power flow with flexible line impedances. First, it is discovered that a flexible-impedance line is equivalent to a constant-impedance line linking a pair of transformers with correlated and continuously adjustable tap ratios. Then, with this circuit equivalent, the original optimization problem is reformulated into a semi-definite pr…
This paper proves that in an unbalanced multi-phase network with a tree topology, the semidefinite programming relaxation of optimal power flow problems is exact when critical buses are not adjacent to each other. Here a critical bus either contributes directly to the cost function or is where an injection constraint is tight at optimality. Our result generalizes a sufficient condition for exact relaxation in single-phase tree networks to tree networks with arbitrary number of phases.
